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Lattices that are on the verge of mechanical instability provide useful models for systems as diverse as architectural structures, crystalline and amorphous solids, sphere packing’s and granular matter, networks of semi-flexible polymers, and protein structure. This talk will explore elastic and mechanical properties and mode structures of model periodic lattices of periodic versions of this lattice, including the origin and nature of zero modes under both periodic (PBC) and free boundary conditions (FBC). It will derive general conditions (a) under which the zero modes under the two boundary conditions are essentially identical and (b) under which phonon modes are gapped with no zero modes in the periodic spectrum but include zero-frequency surface Rayleigh waves in the free spectrum. The gapped states have a topological characterization, similar to that of topological insulators that define the nature of zero-modes at the boundary between systems with different topology.